Date: Feb 10, 2013 6:39 PM Author: Graham Cooper Subject: Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle<br> to Resolve Several Paradoxes On Feb 11, 7:24 am, fom <fomJ...@nyms.net> wrote:

> On 2/10/2013 2:38 AM, Graham Cooper wrote:

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> > On Feb 10, 5:47 pm, fom <fomJ...@nyms.net> wrote:

> >> On 2/9/2013 6:19 PM, Charlie-Boo wrote:

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> >>> On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote:

>

> >>>> How do you see Logic and Set Theory as being the same?

>

> >>> Both are concerned with mappings to {true,false}. A propositional

> >>> calculus proposition is 0-place. A set is 1-place. A relation is any

> >>> number of places. (A relation is a set - of tuples.)

>

> >>> So you have the same rules of inference: Double Negative, DeMorgan

> >>> etc. apply to propositions and sets.

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> >>> To prove incompleteness, Godel had to generalize wffs as expressing

> >>> propositions to expressing sets when the wff has a free variable.

>

> >> Hmm...

>

> >> This is naive set theory (which you have stated

> >> as being fine with your views).

>

> >> I view set theory as being about the existence

> >> of mathematical objects. Naive set theory failed,

> >> in part, because of something in Aristotle--do not

> >> negate "substance". Do not get me wrong. I am

> >> not planning to run out and buy a number 2 while

> >> I pick up my next Turing machine....

>

> >> The problem, however, is that the connection of

> >> mathematics to any metaphysical truth (if such

> >> a statement can be sensible) requires that the

> >> objects represented in physics books (material

> >> objects) correspond with some sort of mathematical

> >> notion. So, while mathematics is abstract,

> >> there must be some sort of interpretation that

> >> accounts for its apparent ability to model

> >> real-world situations.

>

> >> Either physics is a collection of mathematical

> >> hallucinations or there is a better explanation

> >> of set theory.

>

> > Right! the physical world cannot contravene the platonic, so a set of

> > truths may exist and a set of lies not...

>

> > ** in Plato land where (angle1+angle2+angle3=pi) **

>

> > it's the 1 metaphysics principle I subscribe to!

>

> > I think LOGIC is just applying MODUS PONENS.

>

> > backwards to axioms

>

> > a1

> > \

> > theorem ?

> > /

> > a2

>

> > forwards to contradictions

>

> > x

> > /

> > ~theorem

> > \

> > ~x

>

> > Naive set theory should be able to cope with a SUBSET of WFF that have

> > been sieved through various checks. if you can formulate what the

> > real world contradiction is, it can be unstratified.

>

> Herc,

>

> What you say here is Kantian.

>

> Kant called logic the negative criterion

> of truth (forward to contradiction).

>

> And he ascribed the discernment of natural

> laws to presupposition analysis under the

> presumption of causes (backwards to axioms).

>

> For what this is worth, your arguments

> against Cantor's diagonal have been based

> on transversal designs.

>

> The march to infinity is most likely taken

> using finite projective planes described

> by difference sets.

>

> If you want to see why, do an internet

> search on "perfect difference sets" and

> "neighbor detection"

>

> Identity requires infinity. Distinguishability

> in the finitary context of automata is finite.

> With respect to this, equivalence is defined

> negatively. Hence, it presupposes infinity.

>

> Now, identity and diversity are intertwined

> by negation. Leibniz' principle of identity

> of indiscernibles relates an object to all

> of the objects of the system which are not

> the given object. This is like a geometry

> where every pair of points define a line.

>

> Naming is quantization process that requires

> fewer resources. It is Leibniz' principle

> of indiscernibles restricted to "landmarks".

>

> In network analysis they are using perfect

> difference sets for this purpose.

>

> Anyway, it may give you a different

> perspective on some of your thoughts.

>

> Glad you liked the remark.

>

I think NAMING-object and EXISTS(object) and SKOLEM FUNCTIONS are all

part of the same method.

writing a function definition in programming language you just

LET x = 7

all that's required! PUT IT IN WRITING = THEN IT EXISTS(..)

biggerthan(X , Y) <- X = s(Y)

biggerthan(X , 0 ).

Certain Proofs are viable since an Algorithm Exists to find the

required solution.

Extrapoliate that down to all EXIST(X) commands even for a

straightforward object X.

A(Y) E(X) X>Y

/\

| |

\/

A(Y) bigger(Y)>Y

TRUE

X EXISTS because an ALGORITHM is written.

bigger() is the Skolem Function for EXISTS(X)

you just DEFINE in writing what exists!

Then

----------------------------------

AXIOM OF SPECIFICATION gives a NAME to the set IFF it can exist.

EXIST(Y) xeY <-> p(x)

X is just a re-usable programming term like i,k,...

Y is a unique name of the new set, again just by writing it!

----------

I'm currently pondering using NOT(...) my PROLOG theorem type

not( gt (0 , X ))

as I can write that now in PROLOG

<=> ALL(X) ~0>X

and it gives me an ALL(X) Quantifier without breadth first

examination of all records although I think it has to examine inside

the predicate gt(..) coding to return true.

i.e. NOT( NEGATION-AS-FAILURE( gt(0,X) )

would never terminate considering 0, s(0), s(s(0)), ... would all

have to fail a search branch.

However if it was stated not( gt(0,X) )

which atleast MEANS ALL(X) not( gt(0,X) )

then the quantifier syntax could be atomic without any quantifer

predicates added, and humans seem to reason about ALL quantifiers

without working at the ALL RECORDS level... perhaps most PROLOG

recursive programs, usually 2 lines can have a proof by induction

method applied automatically.

Herc

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